A private treaty, also known as a for-sale-by-owner, is in cases where there owner for one reason or another want to sell his / her property. This is a case of willing buyer and willing seller.
A public Auction is a kind of sale where notice of default on homes or property has been filed in the public records. It means the owner has stopped making mortgage payments and the lender has given notice that unless the payments are brought up to date, he / she will sell the property to the highest bidder.
Not all auction and private treaty sales are profitable. One way to make money in real estate is to "buy low and sell high." For auctions and private treaty houses, for a seller to make profits from the sale, they need to couple that principle with fixing up the home or improving it and the amount of profit can be even greater. Besides, many distressed homes fall into disarray and require repairs.